Analysis of non-linear stochastic oscillations by the averaging method

The solution of a quasiconservative non-linear oscillatory system is considered, the right-hand sides of which are proportional to a small parameter. Fundamental relations for solving the problem are obtained by changing to “slow” variables and a combination of the stochastic averaging method and the theory of Markov processes. An efficient numerical algorithm is developed based on the fast Fourier transform that enables the output parameter distribution density and the amplitudes of the oscillations to be obtained. Application of the theory to solve the Duffing–van der Pol equation for an additive and multiplicative stochastic action is considered.